SFP and Projections onto Affine Subspaces
In this chapter we develop and discuss the iteration methods for the solution of the split feasibility problem (SFP) and the computation of metric and Bregman projections onto affine subspaces. At first we examine what operators may be used in the iterative process to handle different kinds of constraints appearing in the SFP. The ones related to constraints in the range of a linear operator depend on a positive parameter which in general has to be chosen a posteriori.
In section 2.2 we show how these parameters can be chosen in case of exact as well as approximate data to ensure convergence of the methods. In case of approximate or noisy data this choice is linked to a discrepancy priniple.
The iteration methods for the SFP are analyzed in section 2.3. They produce sequences which in general have weak accumulation points that are solutions of the SFP. In the following section we show that the same iterative scheme can be used to compute metric and Bregman projections onto affine subspaces that are given via the nullspace or the range of a linear operator. For this case we can even prove strong convergence. In the last two sections we are concerned with possibilities to efficiently implement the methods. We show that the choice of parameters can be replaced by line searches and propose generalized conjugate gradient and sequential subspace methods for the computation of projections onto affine subspaces in case of exact data.
2.1 Convex Constraints and Related Operators
We intend to examine a little more closely the operators we will deal with.
First we recall some facts about linear operators [25, 47]. By L(X, Y ) we denote the Banach space of all continuous linear operators A : X −→ Y endowed with the operator norm
kAk := sup
kxk≤1kAxk . (2.1)
The dual operator A∗∈ L(Y∗, X∗) is defined by
hA∗y∗| xi := hy∗| Axi for all x ∈ X, y∗∈ Y∗ (2.2) and the equalities kA∗k = kAk and N (A∗) =R(A)⊥ are valid. In case X is reflexive we also haveN (A) = R(A∗)⊥ and N (A)⊥ =R(A∗) and in case Y is reflexive we also haveN (A∗)⊥=R(A). An operator A ∈ L(X, Y ) is called compact, if the image A(BX) of the unit ball of X is a relatively compact subset of Y . It is a fact that A is compact iff A∗ is compact and that a com-pact operator A is weak-to-norm-continuous, i.e. if (xn)n is a sequence in X which converges weakly to some x∈ X, then (Axn)nconverges strongly to Ax.
From now on we assume that X is a smooth and uniformly convex Banach space with a (bijective) duality mapping JX with gauge function t 7→ tp−1. If JY is a set-valued duality mapping of another Banach space Y and we write “JY(y)” for some y ∈ Y , then we mean that JY(y) is allowed to be any element in the set JY(y). The additional assumptions in the following definition will be used for the different kinds of contraints in case of exact and approximate data.
Definition 2.1. We call assumption
(C) X is uniformly smooth and a set C∈ C(X) is given.
(A, Q) Given are: a uniformly smooth and uniformly convex Banach space Y with duality mapping JY (with gauge function t7→ tr−1), a compact operator 06= A ∈ L(X, Y ), a set Q ∈ C(Y ) and a con-stant γ∈ (0, 1). The set
MAx∈Q:={x ∈ X | Ax ∈ Q}
is not empty.
(A, y) Given are: an arbitrary Banach space Y with duality mapping JY
(with gauge function t 7→ tr−1), an operator 0 6= A ∈ L(X, Y ), an element y∈ Y and a constant γ ∈ (0, 1). The set
MAx=y :={x ∈ X | Ax = y}
is not empty.
(A, y, +) Given are: an arbitrary Banach lattice Y with positive duality mapping J+, an operator 0 6= A ∈ L(X, Y ), an element y ∈ Y and a constant γ ∈ (0, 1). The set
MAx≤y :={x ∈ X | Ax ≤ y}
is not empty.
(Ci) In addition to assumption (C) a constant β ∈ (0, 1) and convex sets Ci∈ C(X) are given with
dm(C, Ci)≤ ǫmi
and
i→∞lim ǫmi = 0 for all m∈ N .
2.1 Convex Constraints and Related Operators 47 (Aj, Qk) In addition to assumption (A, Q) a constant β ∈ (0, 1), compact operators 06= Aj∈ L(X, Y ) and sets Qk ∈ C(Y ) are given with
kA − Ajk ≤ ηj≤ ηj−1 , dm(Q, Qk)≤ δkm≤ δmk−1
and
j→∞lim ηj = 0 , lim
k→∞δmk = 0 for all m∈ N .
(Aj, yk) In addition to assumption (A, y) a constant β∈ (0, 1), operators 06= Aj∈ L(X, Y ) and elements yk∈ Y are given with
kA − Ajk ≤ ηj ≤ ηj−1 , ky − ykk ≤ δk≤ δk−1
and
j→∞lim ηj= 0 , lim
k→∞δk = 0 .
(Aj, yk, +) In addition to assumption (A, y, +) the same holds as under as-sumption (Aj, yk).
Under assumption (C) we define the operator TC: X−→ X by
TC(x) := ΠCp(x) . (2.3)
Under assumption (A, Q) we define for µ > 0 the operators TA,Q,Πµ , TA,Q,Pµ : X−→ X by
TA,Q,Πµ (x) := JX∗
JX(x)− µA∗
JY(Ax)− JY ΠQr(Ax)
, (2.4) and
TA,Q,Pµ (x) := JX∗
JX(x)− µA∗JY Ax− PQ(Ax)
, (2.5)
whereby Πr is the Bregman projection and P is the metric projection in Y . For Q ={y} with some y ∈ Y under assumption (A, y) we get the (possibly set-valued) operator
TA,yµ : X−→ 2X with TA,{y},Pµ (x) =
TA,yµ (x) := JX∗ JX(x)− µA∗JY(Ax− y) . (2.6) Under assumption (A, y, +) we define for µ > 0 the (possibly set-valued) op-erator
TA,y,+µ : X−→ 2X by
TA,y,+µ (x) := JX∗
JX(x)− µA∗J+ (Ax− y)+
. (2.7)
In Hilbert spaces TA,yµ and TA,Q,Pµ are just the familiar operators TA,yµ (x) = x− µA∗(Ax− y) and TA,Qµ (x) = x− µA∗ Ax− PQ(Ax) , which appear in the ordinary Landweber methods and the CQ algorithm for the SFP. Operator TA,Q,Πµ may also be useful in the context of more general Bregman projections.
For an operator T : X−→ 2X we denote by Fix(T ) :={x ∈ X | x ∈ T (x)}
the set of all fixed points of T and by
S-Fix(T ) :={x ∈ X | x = T (x)}
the set of all strong fixed points of T . Obviously S-Fix(T )⊂ Fix(T ) and if T is single-valued then these sets coincide.
Proposition 2.2.
(a) Under assumption (C) we have
Fix(TC) = C .
(b) Under assumption (A, Q) and for all µ > 0 we have Fix(TA,Q,Πµ ) = Fix(TA,Q,Pµ ) = MAx∈Q. (c) Under assumption (A, y) and for all µ > 0 we have
Fix(TA,yµ ) = S-Fix(TA,yµ ) = MAx=y. (d) Under assumption (A, y, +) and for all µ > 0 we have
Fix(TA,y,+µ ) = S-Fix(TA,y,+µ ) = MAx≤y.
Proof. (a) is just 1.26 (a). For x∈ MAx∈Q we have ΠQr(Ax) = Ax = PQ(Ax) and thus TA,Q,Πµ (x) = x = TA,Q,Pµ (x). Hence MAx∈Q ⊂ Fix(TA,Q,Πµ ) and MAx∈Q ⊂ Fix(TA,Q,Pµ ). Conversely for x∈ Fix(TA,Q,Πµ ) we get
x = TA,Q,Πµ (x)
⇔ JX(x) = JX(x)− µA∗
JY(Ax)− JY ΠQr(Ax)
⇔ A∗
JY(Ax)− JY ΠQr(Ax)
= 0 .
Since MAx∈Q is supposed to be non-empty, we take some z∈ X with Az ∈ Q and get
2.1 Convex Constraints and Related Operators 49 0 =D
A∗
JY(Ax)− JY ΠQr(Ax) x − z
E
=JY(Ax)− JY ΠQr(Ax)
Ax− Az
=JY(Ax)− JY ΠQr(Ax)
Ax− ΠQr(Ax) +JY(Ax)− JY ΠQr(Ax)
ΠQr(Ax)− Az
≥JY(Ax)− JY ΠQr(Ax)
Ax− ΠQr(Ax) ,
because of the validity of the variational inequality (1.30) for ΠQr(Ax) and Az ∈ Q. Since Y is strictly convex by 1.14 (a) the above inequality gives Ax = ΠQr(Ax) ∈ Q. The inclusion Fix(TA,Q,Pµ ) ⊂ MAx∈Q can be shown similarly. In (b) it suffices to show
Fix(TA,yµ )⊂ MAx=y ⊂ S-Fix(TA,yµ ) ,
because S-Fix(TA,yµ )⊂ Fix(TA,yµ ). If x∈ MAx=y then we have Jy(Ax− y) = 0 and it follows that MAx=y ⊂ S-Fix(TA,yµ ). Conversely for x ∈ Fix(TA,yµ ) we find some u∗∈ JY(Ax− y) such that
x = TA,yµ (x) ⇔ JX(x) = JX(x)− µA∗JY(Ax− y) ⇔ A∗JY(Ax− y) = 0 . Since MAx=y is supposed to be non-empty, we take some z∈ X with Az = y and get
0 =hA∗JY(Ax− y) | x − zi = hJY(Ax− y) | Ax − yi = kAx − ykr, which gives Ax = y and thus Fix(TA,yµ )⊂ MAx=y. In (c) it again suffices to show
Fix(TA,y,+µ )⊂ MAx≤y ⊂ S-Fix(TA,y,+µ ) .
If Ax≤ y then we get (Ax−y)+= 0 and thus J+ (Ax−y)+ = 0, which yields TA,y,+µ (x) = x. Hence MAx≤y⊂ S-Fix(TA,y,+µ ). Conversely for x∈ Fix(TA,y,+µ ) we find some u∗∈ J+ (Ax− y)+ such that
x = JX∗ JX(x)− µA∗u∗
⇔ JX(x) = JX(x)− µA∗u∗ ⇔ A∗u∗= 0 . Since MAx≤yis supposed to be non-empty, we find some z∈ X with y−Az ≥ 0 and by the properties of the positive duality mapping 1.21 we get
0 =hA∗u∗| x − zi
=hu∗| Ax − yi + hu∗| y − Azi
=hu∗| (Ax − y)+i + hu∗| y − Azi
=k(Ax − y)+k2+hu∗| y − Azi
≥ k(Ax − y)+k2,
from which we infer that Ax− y ≤ 0 and thus Fix(TA,y,+µ )⊂ MAx≤y. ⊓⊔
The operators are also linked to subdifferentials of certain functionals.
Proposition 2.3. We assume (A, Q), (A, y) or (A, y, +) and accordingly de-fine the functions fA,Q,P, fA,y, fA,y,+: X−→ R by
fA,Q,P(x) := 1
rkAx − PQ(Ax)kr, fA,y(x) := 1
rkAx − ykr, fA,y,+(x) := 1
2k(Ax − y)+k2. Then we have for all x∈ X
A∗JY Ax− PQ(Ax)
⊂ ∂fA,Q,P(x) , A∗JY(Ax− y) ⊂ ∂fA,y(x) , A∗J+ (Ax− y)+
⊂ ∂fA,y,+(x) .
Proof. The assertions for fA,yand fA,y,+follow immediately by 1.12 and 1.22.
We prove the assertion for fA,Q,P. For all x, y ∈ X we get by 1.12 and the variational inequality for the metric projection (1.16)
fA,Q,P(y)− fA,Q,P(x)
= 1
rkAy − PQ(Ay)kr−1
rkAx − PQ(Ax)kr
≥JY Ax− PQ(Ax)
Ay− PQ(Ay) − Ax − PQ(Ax)
=A∗JY Ax− PQ(Ax) y− x +JY Ax− PQ(Ax)
PQ(Ax)− PQ(Ay)
≥A∗JY Ax− PQ(Ax)
y− x .
⊓
⊔
We do not know whether operator TA,Q,Πµ is also linked to a subdifferential of a functional fA,Q,Π. The canonical candidates fA,Q,Π(x) = ∆r Ax, ΠQr(Ax) or fA,Q,Π(x) = ∆r ΠQr(Ax), Ax do not seem to work (do they?). However by 1.24 (e) for fixed z∈ X we have
A∗
JY(Az)− JY ΠQr(Az)
⊂ ∂fA,Q,ΠQr(Az)(z) with
fA,Q,ΠrQ(Az)(x) = ∆r ΠQr(Az), Ax
, x∈ X .